Extracted from the GM's Metaphysic Monograph. A few additional notes have been added in italics.
In the timeless realm before existence, there resided two beings of form nigh incomprehensible to mortal minds, each bent in diametric opposition to the other. Let us call the first Order, and think of it as standing for rules, logic, peace, stillness, and so forth. Let us call its antagonist Chaos, and consider its position one of discord, change, stochasticity, et cetera. Each consisted of what one might approximate to a 'mind' or guiding will, and a physical 'body' or manifestation.
The two battled for an instantaneous eternity, but it was Chaos that struck the final blow, sundering Order into a near-infinite quantity of near-infinitesimal pieces, both in mind and in body.
And so it was that Order won this conflict - for its shattered body became, in an immense explosion detectable throughout the history that would follow, a universe governed by physical laws, predictable changes, and linear time. 1)
The many fragments of Order's mind, meanwhile, became what one might interpret as souls - consciousnesses of free will floating in a dimension 'just off to the side' of material space. In the early aeons of the universe, some of these latched on to physical forms capable of hosting them, and so, on hospitable worlds throughout creation, life came into being. As children of these creatures were born, existing unattached souls could be pulled through into them.
Upon death of living things, however, though their souls are released from their mortal bonds, they find themselves forever changed. The universe, in Order's image, is supposed to be governed by rules, and hence be deterministic, and yet ensouled beings have free will - these do not, strictly, make sense together. As a result, souls find themselves gradually through life scarred by the chaos of their contradiction, leaving them unable to be reborn into a new body after their death.
While Order was reborn as a universe in its own image, Chaos found itself banished to a realm outside of Order's perfect world. It was not giving up, of course, but with its form being so diametrically opposed to the universe, Chaos could not interact with it directly - it was beyond its reach. So Chaos took its own mind, and stripped it from its body, left lifeless in the void. This alone was not enough - to interact with Order's universe, it would need to become a less concentrated force of entropy. So Chaos's mind split itself into four lesser forms, each focused on but a single aspect:
These entropic forms seek to unpick Order's universe, plunging it into chaos. Dominance, Body Horror, and Paranoia roam from place to place, twisting and manipulating life and matter in their own style, for their own pleasure. Vastness, however, much prefers to simply be far away from ordered matter. While out in the depths of nothingness, it discovered a curious artefact of the laws of physics: Black Holes. It transpires that the centre of a black hole simply leads outside of the universe… where, in the void beyond, the abandoned body of Chaos sits. Slowly but surely, Vastness was able to pull some of this chaotic matter through, with which to forge chaotic minions capable of interacting far more directly and intimately with the universe. It has shared this knowledge with its compatriots, and in the subsequent aeons they have pulled through a sufficient quantity of this matter to form a small but formidable army.
The slight chaos inherent in the contradictory nature of ensouled beings has become crucial to how living things and the entropic forms interact. The emotion known as 'fear', present in all living things, is a manifestation of this chaos. The original entity Chaos was beyond mortal comprehension, and so when a living being encounters anything formed from Chaos, they perceive it in a manner warped by this emotion. In turn, the entropic forms have learnt how to abuse this fact, and so it is that, in areas of space within their reach, the fears of its living denizens start to come true…
GM Addendum: The forms the Apparitions took, and how Deepest Darkest Fears came into play, revolved around this. See also: So, how did Deepest Darkest Fears work?.
When an ensouled creature dies, its soul, scarred by the chaos it has accumulated over its life, is separated from its body and moves on. The chaotic energy is left behind, usually not doing much. However, when the entropic forms are in proximity, they can use and reshape this energy to their advantage, strengthening their minions and improving their influence. Essentially, death empowers them.
GM Addendum: Crucially, if a creature died and Apparitions were not already present, there was a very high likelihood that Apparitions would soon appear, and that the chasing fleet led by Dominance would get closer. This added an interesting twist to the numerous PC murder plans that relied on trying to cover up killings by making it look like Apparitions did it2). And yes: if you killed people, you were directly responsible for bringing the Apparitions closer than they would otherwise have been3).
Until recently, the entropic forces had not been near our solar system, and life on Earth had been getting on just fine - albeit with the chaotic energies left behind by the dead just sitting around, not doing much. But now, drawn to this immense source of entropy, they have arrived - and with what is left of humanity fleeing into the stars, they are giving chase.
GM Addendum: From the perspective of at-game-start.
(Summary: Afflicted characters are already 'dead', but have been brought back by the Fears as unwitting sleeper agents.)
When a living being dies, what should happen is its soul departs its body, though the soul is scarred by the chaos of contradiction it has built up over its lifetime, so cannot be reincarnated. However, as they hunt down souls across the universe, the Fears have found a good way to throw a little more chaos into the mix is to have agents unwittingly working on their behalf amongst the flock of souls.
To this end, the Fears very occasionally, just as a creature is dying within their reach, take its just-departed soul, warp it to be under their own subliminal control, and forcibly stuff it back into its body. Besides a slight sense of unease, the afflicted creature is not aware of dying, though will be aware that they almost died.
How this worked mechanically: First, afflicted characters had to die. Some had near-death-experiences already in their backstories, and in these cases, this was taken to be that they had actually died, but thought they had just escaped. For the remainder, they found themselves in near-death situations in the first two turns of the game - either through genuine foolishness, or (more often) us forcing it on them. In this case, players were given the impression that they were lucky to be alive, usually having just been unconscious, but no indication that this was MF4 was given.
After this point, the GMs Apparitions would try to hint at characters to follow a certain path that would lead to treachery. The most common medium for this was dreams. If you did not have MF4, then your dreams were just whatever random stuff your GM could come up with when writing turnsheets! This was a smokescreen for the dreams of characters with MF4, which actually had meaning. Typically, the aim was to raise suspicion of some innocuous thing, to tempt players into messing with it. Other means of manipulation included various forms of hallucinations, such as mishearing radio transmissions, or mis-seeing circuits they were rewiring. Ultimately some characters were easier to manipulate than others, while a handful of the actively malicious characters were very easy to push in a particular direction…
(While non-MF4 dreams were generally meaningless, those few characters who had Interesting Metaphysical Experiences later in the game may have had dreams influenced by those.)